Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks
Abstract
We derive analytical expressions for the generalization performance of kernel regression as a function of the number of training samples using theoretical methods from Gaussian processes and statistical physics. Our expressions apply to wide neural networks due to an equivalence between training them and kernel regression with the Neural Tangent Kernel (NTK). By computing the decomposition of the total generalization error due to different spectral components of the kernel, we identify a new spectral principle: as the size of the training set grows, kernel machines and neural networks fit successively higher spectral modes of the target function. When data are sampled from a uniform distribution on a high-dimensional hypersphere, dot product kernels, including NTK, exhibit learning stages where different frequency modes of the target function are learned. We verify our theory with simulations on synthetic data and MNIST dataset.
Cite
@article{arxiv.2002.02561,
title = {Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks},
author = {Blake Bordelon and Abdulkadir Canatar and Cengiz Pehlevan},
journal= {arXiv preprint arXiv:2002.02561},
year = {2021}
}
Comments
ICML 2020 Update: Updated section on asymptotics generalization error for power law spectra, finding agreement with Spigler, Geiger, Wyart 2019 arXiv:1905.10843. Added a section on Discrete measures and an MNIST Experiment. Eigenvalue problem can be approximated by Kernel PCA. Typo fixed on 2/25/2021